Logical Methods in Computer Science Vol. 9(3:20)2013, pp. 1–19 Submitted Mar. 16, 2013 www.lmcs-online.org Published Sep. 17, 2013 INDUCTION IN ALGEBRA: A FIRST CASE STUDY ∗ PETER SCHUSTER Pure Mathematics, University of Leeds, Leeds LS2 9JT, England e-mail address: [email protected] Abstract. Many a concrete theorem of abstract algebra admits a short and elegant proof by contradiction but with Zorn’s Lemma (ZL). A few of these theorems have recently turned out to follow in a direct and elementary way from the Principle of Open Induction distinguished by Raoult. The ideal objects characteristic of any invocation of ZL are eliminated, and it is made possible to pass from classical to intuitionistic logic. If the theorem has finite input data, then a finite partial order carries the required instance of induction, which thus is constructively provable. A typical example is the well-known theorem “every nonconstant coefficient of an invertible polynomial is nilpotent”. 1. Introduction Many a concrete theorem of abstract algebra admits a short and elegant proof by contra- diction but with Zorn’s Lemma (ZL). A few of these theorems have recently turned out to follow in a direct and elementary way from the Principle of Open Induction (OI) distin- guished by Raoult [42]. A proof of the latter kind may be extracted from a proof of the former sort. If the theorem has finite input data, then a finite partial order carries the required instance of induction, which thus is provable by mathematical induction—or, if the size of the data is fixed, by fully first-order methods. But what is Open Induction? In a nutshell, OI is transfinite induction for subsets of a directed-complete partial order that are open with respect to the Scott topology. While OI was established [42] as a consequence of ZL, by complementation these two principles are actually equivalent [21] with classical logic but in a natural way. Hence OI is the fragment of transfinite induction of which the corresponding minimum principle just is ZL. Our approach is intended as a contribution to a partial realisation in algebra [13] of the revised Hilbert Programme `ala Kreisel and Feferman (see [17] for a recent account including references), and was motivated by related work in infinite combinatorics [8, 11, 14, 42] as well as by the methods of dynamical algebra [16, 39, 51] and formal topology [30, 44, 46]. In Hilbert’s terminology, the “ideal objects” characteristic of any invocation of ZL are eliminated by passing to OI, and it is made possible to work with “finite methods” only, e.g. to pass from classical to intuitionistic logic. 2012 ACM CCS: [Theory of computation]: Logic—Proof theory / Constructive mathematics. Key words and phrases: constructive algebra; Hilbert’s Programme; intuitionistic logic; open induction; Zorn’s Lemma. ∗ This is a revised and extended journal version of the author’s LICS 2012 conference paper [48]. LOGICAL METHODS c P. Schuster Ð IN COMPUTER SCIENCE DOI:10.2168/LMCS-9(3:20)2013 CC Creative Commons 2 P. SCHUSTER A typical example, studied before [41, 43] and taken up in this paper, is the well-known theorem “every nonconstant coefficient of an invertible polynomial is nilpotent”. More formally, this can be put as fg = 1 e (ue = 0) (1.1) →∃ where f and g are polynomials with coefficients in an arbitrary commutative ring n m i j f = X aiT , g = X bjT i=0 i=0 and u = ai0 where 1 i0 n. The customary short and elegant proof of (1.1) works by reduction to the case≤ of polynomials≤ over an integral domain fg = 1 u = 0 → or, equivalently, by reduction modulo any prime ideal P of the given ring: fg = 1 P (u P ) . (1.2) →∀ ∈ This special case is readily settled by looking at the degrees, or more explicitly by a poly- nomial trick due to Gauß [15, 29]. In order to reduce (1.1) to (1.2), it is natural to invoke P (u P ) e (ue = 0) . (1.3) ∀ ∈ →∃ But the latter, a variant of Krull’s Lemma, is normally deduced from ZL by a proof by contradiction, which is anything but an argument using only finite methods. In addition, a universal quantification over prime ideals P occurs, which are ideal objects (see e.g. [17]). These foundational issues aside, there is a practical problem. By decomposing (1.1) into (1.2) and (1.3) one virtually loses the computational information the hypothesis of (1.1) is made of; in particular [41, 43] the proof falls short of being an algorithm for computing an exponent e under which the nilpotent u vanishes. However, we can still extract a proof that is based on induction over a finite partial order; and the proof tree one can grow alongside the induction encodes an algorithm which computes the desired exponent. That our method does work may seem less surprising if one takes into account that the theorem has already seen constructive proofs before [41, 43], and that an entirely down-to- earth proof is possible anyway [4, Chapter 1, Exercise 2]. Needless to say, each of those proofs embodies an algorithm; one of them [41] has even been partially implemented in Agda, a proof assistant based on Martin–L¨of type theory. Just as the proof in [41], our constructive proof is gained from a given classical one, the one by reduction to the case of integral domains we have mentioned above. As compared with [41], we keep somewhat closer to the classical proof. The price we have to pay is that we have to suppose certain decidability hypotheses, which need to—and can—be eliminated afterwards by a variant of the G¨odel–Gentzen and Dragalin–Friedman translations. In [41] a simpler instance of this elimination method is built directly into the proof. To be slightly more specific, we we first turn the indirect proof of (1.3) with ZL into a direct deduction from OI; and then transform the latter into a constructive proof of (1.1) by induction over a finite poset. This is possible because the hypothesis of (1.1)—unlike the one of (1.3)—consists of computationally relevant information about a finite amount of elementary data: of nothing but the finitely many equations a0b0 = 1, a0b1 + a1b0 = 0 , . , anbm = 0 . Heuristics aside, all this makes redundant the reduction, and the prime ideals disappear. INDUCTION IN ALGEBRA: A FIRST CASE STUDY 3 1.1. Preliminaries. 1.1.1. Foundations. The overall framework of this note is constructive algebra `ala Kro- necker and Bishop [31, 35]. Due to the corresponding choice of intuitionistic logic, one or the other assumption needs to be made explicit that would be automatic in classical algebra, by which we mean algebra as carried out within ZFC set theory and thus, in particular, with classical logic. For example, we say that an assertion A is decidable whenever A A holds; and that a subset S of a set T is detachable if t S is decidable for each t T∨¬. As moreover the principle of countable choice will∈ not occur, let alone the one∈ of de- pendent choice, our constructive reasoning can be carried out within (a suitable elementary fragment of) the Constructive Zermelo–Fraenkel Set Theory CZF which Aczel [1, 2, 3] has interpreted within Martin-L¨of’s [34] Intuitionistic Theory of Types. Unlike Friedman’s [19] impredicative Intuitionistic Zermelo–Fraenkel Set Theory IZF, this CZF does not contain the axiom of power set. Hence in CZF an unrestricted quantification over subsets—such as the one crucial for this paper, over all prime ideals of an arbitrary ring—in general is a quantification over the members of a class. 1.1.2. Rings. Throughout this paper, R will denote a commutative ring (with unit). We briefly recall some related concepts [4]. An ideal of R is a subset I that contains 0, is closed under addition, and satisfies s I rs I ∈ → ∈ for all r, s R. We write (S) for the ideal generated by a subset S of R: that is, (S) consists ∈ of the linear combinations r1s1 + . + rnsn of elements s1,...,sn of S with coefficients r1, . , rn from R. A radical ideal of R is an ideal I such that r2 I r I ∈ → ∈ for all r R. The radical ∈ √I = r R : e N (re I) { ∈ ∃ ∈ ∈ } of an ideal I is a radical ideal with I √I. An ideal I is a radical ideal if and only if ⊆ I = √I. The radical √0 of the zero ideal 0 = 0 is the nilradical, and its elements are the nilpotents. { } An ideal P is a prime ideal if 1 / P and ∈ ab P a P b P (1.4) ∈ → ∈ ∨ ∈ for all a, b R. Clearly, every prime ideal is a radical ideal. A ring R is an integral domain—for∈ short, a domain—if 1 =0 in R and 6 ab = 0 a = 0 b = 0 (1.5) → ∨ for all a, b R. A quotient ring R/P is a domain if and only if P is a prime ideal. ∈ 4 P. SCHUSTER 1.1.3. Induction. Let (X, ) be a partial order. We do not specify from the outset whether X is a set in the sense of≤CZF, which in the case of our definite interest will anyway be the case, but during heuristics will depend on the choice of a more generous set theory such as IZF. Unless specified otherwise every quantification over the variables x, x′, y, and z is understood as over the elements of the partial order X under consideration.